Inversion of multicomponent, multiazimuth, walkaway VSP data for the stiffness tensor

Abstract

Vertical seismic profiling (VSP), an established technique, can be used for estimating in‐situ anisotropy that might provide valuable information for characterization of reservoir lithology, fractures, and fluids. The P‐wave slowness components, conventionally measured in multiazimuth, walkaway VSP surveys, allow one to reconstruct some portion of the corresponding slowness surface. A major limitation of this technique is that the P‐wave slowness surface alone does not constrain a number of stiffness coefficients that may be crucial for inferring certain rock properties. Those stiffnesses can be obtained only by combining the measurements of P‐waves with those of S (or PS) modes. Here, we extend the idea of Horne and Leaney, who proved the feasibility of joint inversion of the slowness and polarization vectors of P‐ and SV‐waves for parameters of transversely isotropic media with a vertical symmetry axis (VTI symmetry). We show that there is no need to assume a priori VTI symmetry or any other specific type of anisotropy. Given a sufficient polar and azimuthal coverage of the data, the polarizations and slownesses of P and two split shear (S1 and S2) waves are sufficient for estimating all 21 elastic stiffness coefficients cij that characterize the most general triclinic anisotropy. The inverted stiffnesses themselves indicate whether or not the data can be described by a higher‐symmetry model. We discuss three different scenarios of inverting noise‐contaminated data. First, we assume that the layers are horizontal and laterally homogeneous so that the horizontal slownesses measured at the surface are preserved at the receiver locations. This leads to a linear inversion scheme for the elastic stiffness tensor c. Second, if the S‐wave horizontal slowness at the receiver location is unknown, the elastic tensor c can be estimated in a nonlinear fashion simultaneously with obtaining the horizontal slowness components of S‐waves. The third scenario includes the nonlinear inversion for c using only the vertical slowness components and the polarization vectors of P‐ and S‐waves. We find the inversion to be stable and robust for the first and second scenarios. In contrast, errors in the estimated stiffnesses increase substantially when the horizontal slowness components of both P‐ and S‐waves are unknown. We apply our methodology to a multiazimuth, multicomponent VSP data set acquired in Vacuum field, New Mexico, and show that the medium at the receiver level can be approximated by an azimuthally rotated orthorhombic model.

2-Norm Effective Isotropic Hookean Solids

“…Let us consider the elasticity parameters of a generally anisotropic tensor obtained by Dewangan and Grechka [10] from seismic measurements in New Mexico, …”

Invariant Properties for Finding Distance in Space of Elasticity Tensors

“…Let us consider a weakly anisotropic case, namely, which are the density-scaled elasticity parameters used by Dewangan and Grechka [10]. This is a generally anisotropic medium: it does not exhibit any material symmetry.…”

3C laboratory measurement using laser interferometer

“…This simplifies the estimation of the polarisation, and thus determination of the wave type. There are a number of methods for estimations of anisotropy from seismic field measurements, which utilize wave polarisations (Dewangan and Grechka, 2003). In this paper, we propose a similar method for measuring anisotropy in laboratory experiments.…”